1. science
2. mathematics
3. geometry

Weak solutions for the Ricci flow I
Robert Haslhofer and Aaron Naber∗
April 3, 2015
Abstract
This is the first of a series of papers, where we introduce a new class of estimates for
the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide
a notion of weak solutions to the Ricci flow in the nonsmooth setting. In this first paper,
we prove various new estimates for the Ricci flow, and show that they in fact characterize
solutions of the Ricci flow. Namely, given a family (M, gt )t∈I of Riemannian manifolds, we
consider the path space PM of its space time M = M × I. Our first characterization says that
(M, gt )t∈I evolves by Ricci flow if and only if an infinite dimensional gradient estimate holds
for all functions on PM. We prove additional characterizations in terms of the C 1/2 -regularity
of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral
gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite
dimensional generalizations of much more elementary estimates for the linear heat equation
on (M, gt )t∈I , which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with
lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak
solutions for the Ricci flow. We will develop the structure theory of these weak solutions in
subsequent papers.
Contents
1
Introduction
1.1 Background and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Characterization of supersolutions of the Ricci flow . . . . . . . . . . . . . . . . . . . . . .
1.3 Characterization of solutions of the Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . .
2
Supersolutions of the Ricci flow
∗
2
2
5
6
11
R.H. has been supported by NSF grant DMS-1406394, A.N. has been supported by NSF grant DMS-1406259.
1
3
4
Stochastic calculus on evolving manifolds
3.1 Frame bundle on evolving manifolds . . . . . . .
3.2 Brownian motion and stochastic parallel transport
3.3 Conditional expectation and martingales . . . . .
3.4 Heat equation and Wiener measure . . . . . . . .
3.5 Feynman-Kac formula . . . . . . . . . . . . . .
3.6 Parallel gradient and Malliavin gradient . . . . .
3.7 Ornstein-Uhlenbeck operator . . . . . . . . . . .
Proof of the main theorem
4.1 The gradient estimate . . . . . . . . . .
4.2 Regularity of martingales . . . . . . . .
4.3 Log-Sobolev inequality and spectral gap
4.4 Conclusion of the argument . . . . . . .
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A A variant of Driver’s integration by parts formula
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1.1
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Introduction
Background and overview
The Ricci flow, introduced by Richard Hamilton [Ham82], evolves Riemannian manifolds in time and is
given by the equation
∂t gt = −2Ricgt .
(1.1)
As with all geometric equations, the key to the analysis of (1.1) is to prove estimates that are strong enough
to capture the analytic and geometric behavior. Many of the known estimates for the Ricci flow are similar
in nature to – but often have been harder to develop than – the corresponding estimates for other geometric
equations. Since the geometry itself is evolving, even the most basic geometric quantities, like the heat
kernel, can behave quite badly. Furthermore, many techniques from geometric analysis that rely on the presence of an ambient space (or a fixed underlying manifold) are not available for the Ricci flow. In particular,
it has been a longstanding open problem to find a notion of weak solutions for the Ricci flow.
The goal of this paper, the first in a series, is to introduce a new class of estimates for the Ricci flow.
Our new estimates not only give new information about solutions of the Ricci flow, but are designed to be
sufficiently powerful that they give analytic criteria for determining when a family of Riemannian manifolds
solves the Ricci flow. That is, we will see that if a family (M, gt )t∈I of Riemannian manifolds satisfies the
analytic estimates of this paper, then in fact this family solves (1.1). Such analytic criteria can be used to
define weak solutions and have become of increasing importance in other areas of Ricci curvature, see for
instance [LV09, Stu06, AGS14, Nab13], but have not been available up to now for the Ricci flow itself.
2
We start with the comparably simple task of characterizing supersoluions of the Ricci flow, i.e. families
(M, gt )t∈I such that ∂t gt ≥ −2Ricgt , see Section 1.2 and Section 2. As summarized in Theorem 1.5, supersoluions can be characterized in terms of various estimates for the linear heat equation on (M, gt )t∈I . These
estimates generalize the Bakry-Emery-Ledoux estimates for manifolds with lower Ricci curvature bounds
´
[BE85,
BL06], see also McCann-Topping [MT10]. In particular, one can characterize supersolutions in
terms of a log-Sobolev inequality, and a Poincare-inequality. The log-Sobolev inequality is not the one discovered by Perelman [Per02], but the more recent one from Hein-Naber [HN13].
To characterize solutions of the Ricci flow, and not just supersolutions, we prove infinite dimensional
generalizations of the above estimates. Motivated by work in stochastic analysis [Mal84, Dri92, Fan94,
AE95, Hsu97] and prior work of the second author [Nab13], our approach to finding such infinite dimensional generalizations is to do analysis on path space. More precisely, it turns out that the right path space
to consider, is the space PM of continuous curves in the space-time M = M × I, which are allowed to move
arbitrarily along the manifold M but are required to move backwards along the I factor with unit speed. To
be able to do analysis on PM we have to set up quite a bit of machinery from stochastic analysis, notably the
notions of Wiener measure, stochastic parallel transport, parallel gradient and Malliavin gradient, adapted
to our space-time setting. We describe this briefly in Section 1.3.1 and give a comprehensive treatment in
Section 3. For example, the construction of parallel transport is quite subtle, since almost no curve of Brownian motion is C 1 . Nevertheless, using ideas from Eells-Elworthy-Malliavin [Elw82, Mal97], we can make
sense of parallel transport on space-time for almost every curve of Brownian motion, see Section 3.2.
Having set the stage, let us now discuss our infinite dimensional estimates. Our first characterization in
Section 1.3.2 directly relates solutions of the Ricci flow to gradient estimates on path space. Specifically,
we will see that a family (M, gt )t∈I evolves by Ricci flow if and only if a certain gradient inequality (R2)
holds for all functions on PM. We will see how this directly generalizes the gradient estimate (S2) proved
in Theorem 1.5 for supersolutions. Our second characterization in Section 1.3.3 is in terms of the time
regularity of martingales on path space. Specifically, we will see that martingales F τ on path space satisfy
a precise C 1/2 -H¨older estimate (R3) if and only if the family (M, gt )t∈I evolves by Ricci flow. Our third
characterization in Section 1.3.4 is in terms of an infinite dimensional log-Sobolev inequality (R4), and our
final characterization in Section 1.3.5 is in terms of the corresponding spectral gap (R5). Our characterizations of solutions of the Ricci flow can be thought of as infinite dimensional generalizations of the estimates
for supersolutions. Namely, if we evaluate our infinite dimensional estimates for the simplest possible test
functions, i.e. functions on path space that only depend on the value of the curve at a single time, then we
actually recover the finite dimensional estimates from Theorem 1.5. Of course, there are many more sophisticated test functions that we can plug in our estimates, and this is one of the reasons why our estimates
are actually strong enough to characterize solutions, and not just supersolutions. Our characterizations of
solutions of the Ricci flow constitute the main results of this article and are summarized in Theorem 1.22.
Let us also emphasize that Theorem 1.22 truly relies on ideas from stochastic analysis, i.e. doing analysis
on path space PM, as it seems that analysis on (M, gt )t∈I can only be used to characterize supersolutions but
3
not solutions. In fact, some indications that stochastic analysis might be useful in the study of Ricci flow
have already appeared previously in the literature: Arnoundon-Coulibaly-Thalmaier proved the existence
of Brownian motion in a time dependent setting [ACT08] (see also [Cou11]), and used this to prove a Bismut type formula for the Ricci flow. Kuwada-Philipowski studied the relationship between time dependent
Brownian motion and Perelman’s L-geodesics and obtained a nice nonexplosion result [KP11b, KP11a]
(see also [Che12]), and Guo-Philipowski-Thalmaier found some applications of stochastic analysis to ancient solutions [GPT13]. Based on our new estimates there are many more directions to explore.
In future papers of this series we will use our estimates to investigate singularities in the Ricci flow. In
most situations, the Ricci flow develops singularities in finite time. Typically, the curvature blows up in
certain regions but remains bounded on the remaining parts of the manifold [Ham95]. One would then like
to understand these singularities and find ways to continue the flow beyond the first singular time.
The formation of singularities is of course an ubiquitous phenomenon in the study of nonlinear PDEs.
For other geometric evolution equations there are good notions of weak solutions that allow one to continue the flow through any singularity, e.g. Brakke and level set solutions for the mean curvature flow
[Bra78, ES91, CGG91], and Chen-Struwe solutions for the harmonic map heat flow [CS89]. For the Ricci
flow however, it is only known in a few special - albeit very important - cases, how to continue the flow
through singularities. Most notably, Perelman’s Ricci flow with surgery [Per02, Per03] provides a highly
successful way to deal with the formation of singularities in dimension three. Surgery has also been implemented in the case of four-manifolds with positive isotropic curvature [Ham97, CZ06]. Recently, KleinerLott proved the beautiful result that as the surgery parameters degenerate it is possible to pass to certain
limits, called singular Ricci flows [KL14]. Also, there has been a lot of progress in the K¨ahler case, see e.g.
Song-Tian [ST09] and Eyssidieux-Guedj-Zeriahi [EGZ14]. In most other cases however, it is a widely open
problem how to deal with the formation of singularities.
In the second paper of this series we will use the estimates of this first paper to give a notion of the Ricci
flow for a family of metric-measure spaces. Using analytic characterizations to define weak solutions is a
well developed tool in the context of lower Ricci curvature [LV09, Stu06, AGS14], and more recently in the
context of bounded Ricci curvature [Nab13]. Similarly, based on the characterizations of Theorem 1.22 we
will define a notion of weak solutions for the Ricci flow and develop their theory. We will discuss this in
subsequent papers, but let us briefly describe the idea. We consider metric-measure spaces M equipped with
a time function and a linear heat flow. We call M a weak solution of the Ricci flow if and only if the gradient
estimate (R2) holds on PM. We then establish various geometric and analytic estimates for these weak
solutions. One of our applications concerns a question of Perelman about limits of Ricci flows with surgery
[Per02]. Namely, the metric completion of the space-time of Kleiner-Lott [KL14], which they obtained as a
limit of Ricci flows with surgery where the neck radius is sent to zero, is a weak solution in our sense.
4
1.2
Characterization of supersolutions of the Ricci flow
As a motivation for our approach to characterize solutions of the Ricci flow, let us first characterize supersolutions, i.e. smooth families of Riemannian manifolds such that
∂t gt ≥ −2Ricgt .
(1.2)
To fix notation, let (M, gt )t∈I be a smooth family of Riemannian manifolds, where I = [0, T 1 ]. To avoid
technicalities, we assume throughout the paper that all manifolds are complete and that
sup (|Rm| + |∂t gt | + |∇∂t gt |) < ∞.
(1.3)
M×I
However, all our estimates are independent of the implicit constant in (1.3). We consider the heat equation
(∂t − ∆gt )w = 0 on our evolving manifolds (M, gt )t∈I . For every s, T ∈ I with s ≤ T , and every smooth
function u with compact support, we write P sT u for the solution at time T with initial condition u at time s.
In other words,
Z
(P sT u)(x) =
u(y) H(x, T | y, s)dvolg(s) (y),
(1.4)
M
where H(x, T | y, s) is the heat kernel with pole at (y, s). We write dν(x,T ) (y, s) = H(x, T | y, s)dvolg(s) (y). It is
often useful to think of dν(x,T ) as the adjoint heat kernel measure based at (x, T ).
The following theorem summarizes our characterizations of supersolutions of the Ricci flow.
Theorem 1.5 (Characterization of supersolutions of the Ricci flow). For every smooth family (M, gt )t∈I of
Riemannian manifolds (complete, satisfying (1.3)), the following conditions are equivalent:
(S1) The family (M, gt )t∈I is a supersolution of the Ricci flow,
∂t gt ≥ −2Ricgt .
(S2) For all test functions u, the heat equation on (M, gt )t∈I satisfies the gradient estimate
|∇P sT u| ≤ P sT |∇u|.
(S3) For all test functions u, the heat equation on (M, gt )t∈I satisfies the estimate
|∇P sT u|2 ≤ P sT |∇u|2 .
R
(S4) For all functions u : M → R with M u2 (y) dν(x,T ) (y, s) = 1, we have the log-Sobolev inequality
Z
Z
u2 (y) log u2 (y) dν(x,T ) (y, s) ≤ 4(T − s) |∇u|2gs(y) dν(x,T ) (y, s).
M
M
R
(S5) For all functions u : M → R with M u(y) dν(x,T ) (y, s) = 0, we have the Poincare-inequality
Z
Z
2
u dν(x,T ) (y, s) ≤ 2(T − s) |∇u|2gs dν(x,T ) (y, s).
M
M
5
In essence, this all follows from the Bochner-formula for the heat operator = ∂t − ∆gt ,
|∇u|2 = 2 h∇u, ∇ui − 2|∇2 u|2 − (∂t g + 2Ric)(gradu, gradu),
(1.6)
see Section 2 for the (easy) proof of Theorem 1.5. The reader can also view this as a good toy model for the
more sophisticated infinite-dimensional computations on path space that we carry out in later sections.
Remark 1.7. Theorem 1.5 can be thought of as parabolic version of the Bakry-Emery characterization of
´
nonnegative Ricci curvature [BE85,
BL06]. Another interesting characterization of supersolutions of the
Ricci flow, in terms of the Wasserstein distance, has been given by McCann-Topping [MT10].
1.3
Characterization of solutions of the Ricci flow
In this section we describe our main estimates on path space, and use them to characterize solutions of the
Ricci flow.
1.3.1
Stochastic analysis on evolving manifolds
Our estimates require quite some machinery from stochastic analysis, notably the notions of Wiener measure, stochastic parallel transport, parallel gradient and Malliavin gradient, adapted to our time-dependent
setting. We will now briefly describe these notions, and refer to Section 3 for a more complete treatment.
Let (M, gt )t∈I be a smooth family of Riemannian manifolds, where I = [0, T 1 ]. We recall that we always
assume that our manifolds are complete and that (1.3) is satisfied, though the second assumption is for
convenience. Throughout this work we will think of the evolving manifolds in terms of the space-time
M = M × I. As observed by Hamilton [Ham93] there is a natural space-time connection defined by
g
∇X Y = ∇Xt Y,
1
∇t Y = ∂t Y + ∂t gt (Y, ·)]gt .
2
(1.8)
The point is that this connection is compatible with the metric, i.e. dtd |Y|2gt = 2hY, ∇t Yi.
It is useful to consider space-time curves going backwards in time, c.f. [LY86, Per02]. Namely, for
each (x, T ) ∈ M, we consider the based path space P(x,T ) M consisting of all space-time curves of the form
{γτ = (xτ , T − τ)}τ∈[0,T ] , where {xτ }τ∈[0,T ] is a continuous curve in M with x0 = x.
We equip the path space P(x,T ) M with a probability measure Γ(x,T ) , that we call the Wiener measure
of Brownian motion on our evolving family of manifolds, based at (x, T ). The measure Γ(x,T ) is uniquely
characterized by the following property. If eσ : P(x,T ) M → M k , γ 7→ (xσ1 , . . . , xσk ), is the evaluation map
at σ = {0 ≤ σ1 ≤ . . . ≤ σk ≤ T }, and if we write si = T − σi , then
eσ,∗ dΓ(x,T ) (y1 , . . . , yk ) = H(x, T |y1 , s1 )dvolgs1 (y1 ) · · · H(yk−1 , sk−1 |yk , sk )dvolgsk (yk ),
(1.9)
where H is the heat kernel of ∂t − ∆gt ; see Section 3.2 for the construction of Brownian motion.
It is often convenient to consider the total path space PT M = ∪ x∈M P(x,T ) M. Note that we can identify
(PT M, Γ(x,T ) ) with (P(x,T ) M, Γ(x,T ) ), since the measure Γ(x,T ) concentrates on curves starting at (x, T ). SomeR
times it is also useful to equip the total path space PT M with the measure ΓT = Γ(x,T ) dvolgT (x).
6
The space (PT M, Γ(x,T ) ) can be equipped with a notion of stochastic parallel transport, a family of
isometries Pτ (γ) : (T xτ M, gT −τ ) → (T x M, gT ). If the curves γ were C 1 , then Pτ (γ) would just be the parallel
transport from differential geometry, with respect to the natural space-time connection defined in (1.8). Of
course, almost no curve of Brownian motion is C 1 . Nevertheless, using deep ideas from Eells-ElworthyMalliavin we can still make sense of Pτ (γ) for almost every curve γ, see Section 3.2 for the construction.
The space (PT M, Γ(x,T ) ) can be equipped with two natural notions of gradient. Suppose first that F :
P(x,T ) M → R is a cylinder function, i.e. a function of the form F = u ◦ eσ , where eσ : P(x,T ) M → M k
is an evaluation map and u : M k → R is a smooth function with compact support. If v ∈ (T x M, gT ), then
for almost every (a.e.) curve γ, we can consider the continuous vector field V = {Vτ = P−1
τ v}τ∈[0,T ] along γ,
where Pτ = Pτ (γ) denotes stochastic parallel transport as in the previous paragraph. Note that the directional
derivative DV F(γ) is well defined, as a limit of difference quotients as usual.
The parallel gradient ∇k F(γ) ∈ (T x M, gT ) is then defined by the condition that
DV F(γ) = h∇k F(γ), vi(T x M,gT )
(1.10)
for all v ∈ (T x M, gT ), where V = {Vτ = P−1
τ v}τ∈[0,T ] is the parallel vector field associated to v, as above.
More generally, there is a one parameter family of parallel gradients ∇kσ (0 ≤ σ ≤ T ), which captures the
part of the gradient coming from the time interval [σ, T ]. In particular, ∇k = ∇k0 .
The Malliavin gradient ∇H F is defined along similar lines, but takes values in an infinite dimensional
Hilbert space. Namely, let H be the Hilbert-space of H 1 -curves {vτ }τ∈[0,T ] in (T x M, gT ) with v0 = 0, equipped
RT
with the inner product hv, wiH = 0 h˙vτ , w˙ τ i(T x M,gT ) dτ. Then ∇H F : P(x,T ) M → H is the unique almost
everywhere defined function such that
DV F(γ) = h∇H F(γ), viH ,
(1.11)
for a.e. curve γ, and every v ∈ H, where V = {P−1
τ vτ }τ∈[0,T ] .
Having defined them on cylinder functions, the (σ-)parallel gradient and the Malliavin gradient can be
extended to closed unbounded operators on L2 , see Section 3.6 for details.
Finally, the Ornstein-Uhlenbeck operator L = ∇H∗ ∇H is defined by composing the Malliavin gradient
with its adjoint. More generally, there is a family of Ornstein-Uhlenbeck operators Lτ1 ,τ2 (0 ≤ τ1 < τ2 ≤ T ),
which captures the part of the Laplacian coming from the time interval [τ1 , τ2 ]. In particular, L = L0,T .
1.3.2
Ricci flow and the gradient estimate
Our first characterization of solutions of the Ricci flow is in terms of an infinite dimensional gradient estimate on the associated path space. Let (M, gt )t∈I be smooth family of Riemannian manifolds and let PT M
be its path space, equipped with the Wiener measure and the parallel gradient. If F : PT M → R is a sufficiently nice function, for instance a cylinder function, one can ask whether one can control the gradient of
R
FdΓ(x,T ) viewed as a function of x ∈ M, in terms of some natural gradient of F viewed as a function
PT M
on path space. In fact, the answer to this question turns out to be highly relevant, in that it yields our first
characterization of solutions of the Ricci flow. Namely, we prove that (M, gt )t∈I evolves by Ricci flow if and
7
only if the gradient estimate
Z
(R2)
|∇ x
Z
PT M
FdΓ(x,T ) | ≤
PT M
|∇k F|dΓ(x,T ) ,
holds for all function F ∈ L2 (PT M, ΓT ) (for a.e. (x, T ) ∈ M).
Remark 1.12. The infinite dimensional gradient estimate (R2) can be thought of as (vast) generalization of
the finite dimensional gradient estimate (S2) for the heat equation. Namely, let F = u◦eσ : PT M → M → R
be a 1-point cylinder function, and write s = T − σ. By equation (1.9) the pushforward measure
eσ,∗ dΓ(x,T ) = dν(x,T ) (·, s)
is given by the heat kernel measure dν(x,T ) (y, s) = H(x, T | y, s)dvolg(s) (y), and thus
Z
Z
FdΓ(x,T ) =
u eσ,∗ dΓ(x,T ) = (P sT u)(x).
PT M
(1.13)
(1.14)
M
Moreover, using (1.10) on sees that |∇k F|(γ) = |∇u|gs(eσ (γ)), which together with (1.13) implies that
Z
Z
k
|∇ F|dΓ(x,T ) =
|∇u| eσ,∗ dΓ(x,T ) = (P sT |∇u|)(x).
(1.15)
PT M
M
Thus, in the special case of 1-point cylinder function the estimate (R2) reduces to the finite dimensional heat
equation estimate
(S2)
|∇P sT u| ≤ P sT |∇u|.
Of course, there are many more test functions on path space than just 1-point cylinder function. This is one
of the reasons why our infinite dimensional estimate (R2) is strong enough to characterize solutions of the
Ricci flow, while the finite dimensional heat equation estimate (S2) just characterizes supersolutions.
1.3.3
Ricci flow and the regularity of martingales
Our second characterization of solutions of the Ricci flow is in terms of the regularity of martingales on its
path space. Let (M, gt )t∈I be a smooth family of Riemannian manifolds, and let PT M be its path space. For
every function F ∈ L2 (PT M, Γ(x,T ) ), we can consider the induced martingale {F τ }τ∈[0,T ] ,
Z
τ
F (γ) =
F(γ|[0,τ] ∗ γ0 )dΓγτ (γ0 ),
(1.16)
PT −τ M
where the integral is over all Brownian curves γ0 based at γτ , and ∗ denotes concatenation. The family
0
{F τ }τ∈[0,T ] indeed has the martingale property (F τ )τ = F τ (τ0 ≥ τ) and captures how F depends on the
[0, τ]-part of the curves, see Section 3.3. The quadratic variation [F • ]τ of the martingale {F τ }τ∈[0,T ] is
P
defined by [F • ]τ = lim||{τ j }||→0 k (F τk − F τk−1 )2 , where the limit is taken in probability, over all partions
{τ j } of [0, τ] with mesh going to zero, see Section 3.3. It turns out that solutions of the Ricci flow can be
•
characterized in terms of certain bounds for d[Fdτ ]τ . Namely, we prove that (M, gt )t∈I evolves by Ricci flow if
and only if the estimate
Z
Z
d[F • ]τ
(R3)
dΓ(x,T ) ≤ 2
|∇kτ F|2 dΓ(x,T )
dτ
PT M
PT M
holds for every F ∈ L2 (PT M, Γ(x,T ) ) (for all (x, T ) ∈ M).
8
Remark 1.17. The estimate (R3) is a (vast) generalization of (S3). Namely, let F = u◦eσ : PT M → M → R
be a 1-point cylinder function, and write s = T − σ. If ε > 0, then by (1.13) and (1.16) we have
Z
ε
F (γ) =
u(y)dνγε (y, s) = (P s,T −ε u)(xε ).
(1.18)
M
Appying this twice and using the short time asymptotics of the heat kernel, one can compute that
Z
Z
2
d[F • ]τ
1
|τ=0 dΓ(x,T ) = lim
F ε − (F ε )0 dΓ(x,T )
ε→0 ε PT M
dτ
PT M
!2
Z
Z
1
= lim
(P s,T −ε u)(z) −
(P s,T −ε u)(ˆz) dν(x,T ) (ˆz, T − ε) dν(x,T ) (z, T − ε) = 2|∇P sT u|2 (x).
ε→0 ε M
M
Thus, in the special case of 1-point cylinder functions, (R3) for τ = 0 reduces to the estimate1
(S3)
1.3.4
|∇P sT u|2 ≤ P sT |∇u|2 .
Ricci flow and the log-Sobolev inequality
Our third characterization of solutions of the Ricci flow is in terms of a log-Sobolev inequality on its path
space. Log-Sobolev inequalities have a long history, going back to Gross [Gro75]. In the context of Ricci
flow, they appear in Perelman’s monotonicity formula [Per02] and also in the inequality (S4) of HeinNaber [HN13]. We characterize solutions of the Ricci flow via an infinite dimensional generalization of
the inequality (S4). Namely, we prove that (M, gt )t∈I evolves by Ricci flow if and only if the log-Sobolev
inequality
Z
Z
(F 2 )τ2 log (F 2 )τ2 − (F 2 )τ1 log (F 2 )τ1 dΓ(x,T ) ≤ 4
hF, Lτ1 ,τ2 FidΓ(x,T ) ,
(R4)
PT M
PT M
holds for every F in the domain of the Ornstein-Uhlenbeck operator Lτ1 ,τ2 (for all (x, T ) ∈ M and all
0 ≤ τ1 < τ2 ≤ T ). Here, (F 2 )τ denotes the martingale induced by F 2 .
Remark 1.19. If τ1 = 0 and τ2 = T the inequality (R4) takes the somewhat simpler form
Z
Z
2
2
F log F dΓ(x,T ) ≤ 4
|∇H F|2 dΓ(x,T )
PT M
PT M
(1.20)
R
for all F with P M F 2 = 1. Specializing further, for a 1-point cylinder function F = u◦eσ : PT M → M → R
T
(s = T − σ), using (1.11) one can see that |∇H F|2H (γ) = (T − s)|∇u|gs(eσ (γ)), c.f. Proposition 3.52. Together
with (1.13) this shows that (R4) then reduces to (S4).
1.3.5
Ricci flow and the spectral gap
Our final characterization of solutions of the Ricci flow is in terms of the spectral gap of the OrnsteinUhlenbeck operator on its path space.2 We prove that (M, gt )t∈I evolves by Ricci flow if and only if the
For τ , 0, one gets the estimate PtT |∇P st u|2 ≤ P sT |∇u|2 , which is easily seen to be equivalent to (S3).
It is of course well known that a log-Sobolev inequality implies a spectral gap. However, the important point we
prove is that the spectral gap is in fact strong enough to characterize solutions of the Ricci flow.
1
2
9
Ornstein-Uhlenbeck operator Lτ1 ,τ2 (for all (x, T ) ∈ M and all 0 ≤ τ1 < τ2 ≤ T ) satisfy the spectral gap
estimate
Z
Z
(R5)
(F τ2 − F τ1 )2 dΓ(x,T ) ≤ 2
hF, Lτ1 ,τ2 FidΓ(x,T ) .
PT M
PT M
Remark 1.21. In the special case of 1-point cylinder functions, the estimate (R5) again reduces to (S5).
1.3.6
Summary of main results
Our main results are summarized in the following theorem.
Theorem 1.22 (Characterization of solutions of the Ricci flow). For every smooth family (M, gt )t∈I of Riemannian manifolds (complete, satisfying (1.3)), the following conditions are equivalent:
(R1) The family (M, gt )t∈I evolves by Ricci flow,
∂t gt = −2Ricgt .
(R2) For every F ∈ L2 (PT M, ΓT ), we have the gradient estimate
Z
Z
|∇ x
FdΓ(x,T ) | ≤
|∇k F|dΓ(x,T ) .
PT M
PT M
(R3) For every F ∈ L2 (PT M, Γ(x,T ) ), the induced martingale {F τ }τ∈[0,T ] satisfies the estimate
Z
Z
d[F • ]τ
dΓ(x,T ) ≤ 2
|∇kτ F|2 dΓ(x,T ) .
dτ
PT M
PT M
(R4) The Ornstein-Uhlenbeck operator Lτ1 ,τ2 on based path space L2 (PT M, Γ(x,T ) ) satisfies the log-Sobolev
inequality
Z
Z
2 τ2
2 τ2
2 τ1
2 τ1
(F ) log (F ) − (F ) log (F ) dΓ(x,T ) ≤ 4
hF, Lτ1 ,τ2 Fi dΓ(x,T ) .
PT M
PT M
(R5) The Ornstein-Uhlenbeck operator Lτ1 ,τ2 on based path space L2 (PT M, Γ(x,T ) ) satisfies the spectral
gap estimate
Z
Z
τ2
τ1 2
hF, Lτ1 ,τ2 FidΓ(x,T ) .
(F − F ) dΓ(x,T ) ≤ 2
PT M
PT M
Remark 1.23. As explained above, in the special case of 1-point cylinder functions the estimates (R2)–(R5)
reduce to the estimates (S2)–(S5), respectively.
Remark 1.24. Further characterizations are possible. In particular, we have an L2 -version of the gradient
estimate, and a pointwise L1 -version of the martingale estimate, see (R2’) and (R3’) in Section 4.
Outline. This article is organized as follows. In Section 2, as a warmup for the proof of the main theorem, we
prove Theorem 1.5 characterizing supersolutions of the Ricci flow. In Section 3, we set up the machinery of
stochastic analysis in our setting of evolving manifolds. In Section 4, we prove the main theorem (Theorem
1.22) characterizing solutions of the Ricci flow.
10
2
Supersolutions of the Ricci flow
In this short section we prove Theorem 1.5, characterizing supersolutions of the Ricci flow
Proof of Theorem 1.5. We will prove the implications (S3)⇔(S1)⇔(S2) and (S1)⇒(S4)⇒(S5)⇒(S3).
(S1)⇔(S3): If gt is a supersolution of the Ricci flow, then the Bochner formula (1.6) implies
|∇P st u|2 ≤ 0.
(2.1)
Thus, |∇P st u|2 − P st |∇u|2 is a subsolution of the heat equation. Since it is zero for t = s, it stays nonpositive
for all t > s, in particular |∇P sT u|2 ≤ P sT |∇u|2 . To prove the converse implication, assume that (∂t g +
2Ric)(X, X) < 0 for some unit tangent vector X ∈ T x M at some time s. Choose a test function u with
∇u(p) = X and ∇2 u(p) = 0. Then by (1.6) we have ∂t |∇P st u|2 > ∆|∇u|2 at p at t = s; this contradicts (S3).
(S1)⇔(S2): If gt is a supersolution of the Ricci flow, then using the Bochner formula (1.6) and the
Cauchy-Schwarz inequality we obtain
!
1
1 |∇|∇P st u|2 |2
1
2
|∇P st u| +
|∇P st u| =
≤ 0.
(2.2)
|∇P st u| 2
4 |∇P st u|2
Thus, |∇P st u| − P st |∇u| is a subsolution of the heat equation. Since it is zero for t = s, it stays nonpositive for
all t > s, in particular |∇P sT u| ≤ P sT |∇u|. The converse implication follows by considering a test function
as above.
(S1)⇒(S4): Let w > 0. We start by deriving another estimate for the heat equation. Using the Bochner
formula (1.6) and the Peter-Paul inequality we compute
D
E
!
∇|∇P st w|2 , ∇P st w
|∇P st w|2
|∇P st w|2
|∇P st w|4
=
+2
−
2
≤ 0.
(2.3)
P st w
P st w
(P st w)2
(P st w)3
2
2
|∇w|
st w|
Thus, |∇P
P st w − P st w is a subsolution of the heat equation. Since it is zero for t = s, this implies the
estimate
|∇P sr w|2
|∇w|2
≤ P sr
.
(2.4)
P sr w
w
Now, using the heat kernel homotopy principle [HN13, (3.7)] and (2.4) we compute
!
! Z T
!
Z
Z
Z
Z
|∇P sr w|2
|∇w|2
PrT
w log w dν −
w dν log
w dν =
(x) dr ≤ (T − s)
dν.
P sr w
w
s
(2.5)
Substituting w = u2 this implies the log-Sobolev inequality (S4).
R
(S4)⇒(S5): This follows by evaluating (S4) for w2 = 1 + εu with u dν = 0.
(S5)⇒(S3) By the heat kernel homotopy principle [HN13, (3.7)] we have
Z
!2
Z
2
u dν −
udν
=2
T
Z
PrT |∇P sr u|2 (x) dr.
(2.6)
s
Thus, if (S3) fails at some (x, T ), then (S5) fails for dν(x,T ) with |T − s| small enough.
11
3
Stochastic calculus on evolving manifolds
We will now discuss in more detail the required background from stochastic analysis, adapted to our timedependent setting. There are numerous excellent references for stochastic analysis on manfolds, e.g. [Elw82,
´
Eme89,
Hsu02, IW81, Mal97, Str00]. For readers who wish to focus on one single reference which is
particularly close in spirit to the content of the present section we recommend the book by Hsu [Hsu02].
3.1
Frame bundle on evolving manifolds
To set things up efficiently, we will first explain how to formulate the differential geometry of evolving
manifolds in terms of the frame bundle. For the frame bundle formalism in the time-independent case, see
e.g. Kobayashi-Nomizu [KN96], for the frame bundle formalism for the Ricci flow, see Hamilton [Ham93].
Let (M, gt )t∈I , I = [0, T 1 ], be a smooth family of Riemannian manifolds, and write M = M × I. Let Y
be a time dependent vector field. For each X ∈ (T x M, gt ) we can compute the covariant spatial derivative
g
∇X Y = ∇Xt Y using the Levi-Civita connection of the metric gt . The covariant time derivative is defined as
∇t Y = ∂t Y + 21 ∂t gt (Y, ·)]gt . The point is that this gives metric compatibility, namely dtd |Y|2gt = 2hY, ∇t Yi.
Consider the On -bundle π : F → M, where the fibres F(x,t) are given by the orthogonal maps u : Rn →
(T x M, gt ), and g ∈ On acts from the right via composition. The horizontal lift of a curve γt in M is a curve
ut in F with πut = γt such that ∇γ˙ t (ut e) = 0 for all e ∈ Rn . Given a vector αX + β∂t ∈ T (x,t) M and a frame
u ∈ F(x,t) there is a unique horizontal lift αX ∗ +βDt with π∗ (αX ∗ +βDt ) = X. Here, X ∗ is just the horizontal lift
d
|0 u s , where u s is the horizontal lift based at u of
of X ∈ T x M with respect to the fixed metric gt , and Dt = ds
the curve s 7→ (x, t+ s) with x constant. Most of the time we only consider curves of the form γτ = (xτ , T −τ).
We denote space-time parallel transport by Pτ1 ,τ2 = uτ2 u−1
τ1 : (T xτ1 M, gT −τ1 ) → (T xτ2 M, gT −τ2 ), and observe
that this induces parallel translation maps for arbitrary tensor fields. We write Dτ = −Dt .
Given a representation ρ of On on some vector space V and an equivariant map from F to V, we get a
section of the associated vector bundle F ×ρ V, and vice versa. For example, a time-dependent function f
corresponds to the invariant function f˜ = f π : F → R, and a time-dependent vector field Y corresponds
˜
˜
˜
to a function Y˜ : F → Rn via Y(u)
= u−1 Yπu , which is equivariant in the sense that Y(ug)
= g−1 Y(u).
The
following lemma shows how to compute derivatives in terms of the frame bundle.
∗ ˜ and ∇
g
g
ff = X ∗ f˜, ∂
˜ g
˜
Lemma 3.1 (First derivatives). X
t f = Dt f , ∇X Y = X Y,
t Y = Dt Y.
Proof. The first two formulas are obvious, since the horizontal lift of a function is constant in fibre direction.
To prove the last formula, let ut be a horizontal curve with πut = γt = (x, t), where x is fixed. Then
−1 d
g
˜ t ) = d |t1 u−1
˜ ut = d |t1 Y(u
|0 P−1 Y(x,t1 +s) = u−1
(Dt Y)
t Yπut = ut1
t1 (∇t Y)(x,t1 ) = (∇t Y)u1 .
1
dt
dt
ds t1 ,t1 +s
(3.2)
The third formula follows from a similar computation. In fact, it is a well known formula from differential
geometry with respect to a fixed metric gt .
Let e1 , . . . , en be the standard basis of Rn . We write Hi for the horizontal vector fields Hi (u) = (uei )∗ ,
P
where ∗ denotes the horizontal lift, as before. The horizontal Laplacian is defined by ∆H = ni=1 Hi2 .
12
f = ∆H Y.
ff = ∆H f˜, ∆Y
˜
Lemma 3.3 (Laplacian). ∆
Proof. This is a classical fact from differential geometry with respect to a fixed metric gt .
We also need the notion of the antidevelopment of a horizontal curve (this concept is also known as
Cartan’s rolling without slipping), see e.g. [KN96], generalized to the time-dependent setting. The point is
that the horizontal vector fields provide a way to identify curves in Rn with horizontal curves in F.
Definition 3.4 (Antidevelopment). If {uτ }τ∈[0,T ] is a horizontal curve in F with π(uτ ) = (xτ , T − τ), its
antidevelopment {wτ }τ∈[0,T ] is the curve in Rn that satisfies
dwi
duτ
= Dτ + Hi (uτ ) τ ,
dτ
dτ
3.2
w0 = 0.
(3.5)
Brownian motion and stochastic parallel transport
The goal of this section is to generalize the Eells-Elworthy-Malliavin construction of Brownian motion and
stochastic parallel translation, see e.g. [Hsu02], to our setting of evolving manifolds. We note that a related
construction in the time-dependent setting has been given by Arnoudon-Coulibaly-Thalmaier [ACT08].
The idea is to solve (3.5) in a stochastic setting. This provides a way to identify Brownian curves
{wτ }τ∈[0,T ] in Rn with horizontal Brownian curves {uτ }τ∈[0,T ] in F. The virtue of this approach is that it
yields both Brownian motion on M, via projecting, and stochastic parallel transport, via Pτ1 ,τ2 = uτ2 u−1
τ1 .
Let (M, gt )t∈I , I = [0, T 1 ], be a one-parameter family of Riemannian manifolds, and let π : F → M × I be
the time dependent On -bundle introduced in the previous section. We fix a frame u ∈ F, write π(u) = (x, T ),
and denote the projections to space and time by π1 : F → M and π2 : F → I, respectively. It will be
convenient to work with the backwards time τ, defined by t = T − τ. As before, we write Dτ = −Dt .
Motivated by (3.5), we consider the following stochastic differential equation (SDE) on F:
dUτ = Dτ dτ + Hi (Uτ ) ◦ dW iτ ,
U0 = u.
(3.6)
Here, Wτ is Brownian motion on Rn , and ◦ indicates that the equation is in the Stratonovich sense. To keep
the factor 2 in Hamilton’s Ricci flow, ∂t gt = −2Ricgt , we use the convention that dWτ doesn’t have the
√
j
standard normalization from stochastic calculus, but is scaled by a factor 2, i.e. dWτi dWτ = 2δi j dτ.
Proposition 3.7 (Existence, uniqueness, and Ito formula). The SDE (3.6) has a unique solution {Uτ }τ∈[0,T ] .
The solution satisfies π2 (Uτ ) = T − τ, and does not explode. Moreover, Uτ (ω) is continuous in τ for almost
every Brownian path ω ∈ C([0, T ], Rn ), and for any C 2 -function f : F → R we have the Ito formula
d f (Uτ ) = Hi f (Uτ )dWτi + Dτ f (Uτ )dτ + Hi Hi f (Uτ )dτ.
(3.8)
Proof. We recall that SDEs on manifolds can be reduced to SDEs on Euclidean space, see e.g. [Hsu02, Sec.
1.2]. Choose an embedding F ⊂ RN and suitable extensions of all functions to RN . By the standard theory
of SDEs on Euclidean space, there is a unique solution of the system (a = 1, . . . , N):
dUτa = Daτ dτ + Hia (Uτ ) ◦ dW iτ ,
13
U0 = u.
(3.9)
It follows from a Gronwall type argument that the solution actually stays inside F, see e.g. [Hsu02, Prop.
1.2.8]. This proves existence of a solution of (3.6). Moreover, it is also easy to derive a uniqueness result
for solutions of (3.6) from the standard uniqueness result for SDEs on Euclidean space, see e.g. [Hsu02,
Thm. 1.2.9]. In particular, the solution is independent of the choices of embedding and extensions. Since
Brownian motion in Rn is continuous in τ for almost every path, the same is true for Uτ .
To prove (3.8), we first convert (3.9) into a SDE in the Ito sense. Computationally this is done by dropping
the ◦ and adding one half times the quadratic variation of H(Uτ ) and Wτ :
dUτa = Daτ dτ + Hia (Uτ )dW iτ + 12 dHia (Uτ )dWτi ,
U0 = u.
(3.10)
Now, using Ito calculus in Euclidean space we compute
dHia (Uτ )dWτi = ∂b Hia (Uτ )dUτb dWτi = 2∂b Hia (Uτ )Hib (Uτ )dτ,
(3.11)
and
d f (Uτ ) =∂a f (Uτ )dUτa + 12 ∂a ∂b f (Uτ )dUτa dUτb
=∂a f (Uτ )Daτ dτ + ∂a f (Uτ )Hia (Uτ )dW iτ
+ ∂a f (Uτ )∂b Hia (Uτ )Hib (Uτ ) + ∂a ∂b f (Uτ )Hia (Uτ )Hib (Uτ ) dτ.
(3.12)
Observing that the term in brackets is equal to Hi Hi f (Uτ ), this proves (3.8).
By assumption (1.3) the metrics are equivalent at all times and there exists a distance-like function, i.e. a
smooth function r : M → R such that, after fixing an arbitrary point and o ∈ M,
C −1 (1 + dt (x, o)) ≤ r(x) ≤ C(1 + dt (x, o)),
|∇r| ≤ C,
∇∇r ≤ C
(3.13)
for some C < ∞. Let r˜ : F → R be the extension of r, that is independent of time and the fibre coordinates.
Applying the Ito formula (3.8) to r˜, we see that the solution of (3.9) does not explode, i.e. that Uτ does not
escape to spatial infinity. Finally, for f = π2 the Ito formula takes the simple form dπ2 (Uτ ) = −dτ. Together
with π2 (U0 ) = T , this implies that π2 (Uτ ) = T − τ.
Using Propositon 3.7 we can now define Brownian motion and stochastic parallel transport on our evolving family of Riemannian manifolds.
Definition 3.14 (Brownian motion). We call π(Uτ ) = (Xτ , T − τ) Brownian motion based at (x, T ).
Definition 3.15 (Stochastic parallel transport). The family of isometries Pτ = U0 Uτ−1 : (T Xτ M, gT −τ ) →
(T x M, gT ), depending on τ and the Brownian curve, is called stochastic parallel transport.
Brownian motion comes naturally with its path space, diffusion measure, and filtered σ-algebra.
Definition 3.16 (Based path spaces). We let P0 Rn be the space of continuous curves {ωτ }τ∈[0,T ] in Rn with
ω0 = 0, let Pu F be the space of continuous curves {uτ }τ∈[0,T ] in F with u0 = u and π2 (uτ ) = T − τ, and let
P(x,T ) M be the space of continuous curves {γτ = (xτ , T − τ)}τ∈[0,T ] in M with γ0 = (x, T ).
14
To introduce the diffusion measure, note that Proposition 3.7 defines a map U : P0 Rn → Pu F, U(ω)(τ) =
Uτ (ω). We also have a natural map Π : Pu F → P(x,T ) M, induced by the projection π : F → M × I.
Definition 3.17 (Diffusion measures). Let Γ0 be the Wiener measure on P0 Rn , let Γu = U∗ Γ0 be the probability measure on Pu F obtained by pushing forward via U, and let Γ(x,T ) = (Π ◦ U)∗ Γ0 be the probability
measure on P(x,T ) M obtained by pushing forward via Π ◦ U.
Finally, recall the Wiener space P0 Rn comes naturally equipped with a filtered family of σ-algebras
Στ = Στ (P0 Rn ), which is generated by the evaluation maps eτ1 : P0 Rn → Rn , eτ1 (ω) = ωτ1 with τ1 ≤ τ.
Definition 3.18 (Filtered σ-algebras). We denote by Στ (Pu F) and Στ (P(x,T ) M) (or simply by Στ if there is
no risk of confusion) the pushforward of Στ (P0 Rn ) under the maps U and Π ◦ U, respectively.
3.3
Conditional expectation and martingales
R
If F : Pu F → R is integrable, we write Eu [F] = P F FdΓu for its expectation. More generally, if σ ∈ [0, T ],
u
we write F σ = Eu [F|Σσ ] for the conditional expectation given the σ-algebra Σσ (see Definition 3.18).
R
We recall that the conditional expectation F σ is the unique Σσ -measurable function such that Ω F σ dΓu =
R
FdΓu for all Σσ -measurable sets Ω. Similarly, if F is an integrable function on P(x,T ) M, we also write
Ω
E(x,T ) [F] and F σ = E(x,T ) [F|Σσ ] for its expectation and conditional expectation, respectively.
Proposition 3.19 (Conditional expectation). If F : P(x,T ) M → R is integrable and σ ∈ [0, T ], then for a.e.
Brownian curve {γτ }τ∈[0,T ] the conditional expectation F σ = E(x,T ) [F|Σσ ] is given by the formula
Z
σ
F (γ) =
F(γ|[0,σ] ∗ γ0 ) dΓγσ (γ0 ),
(3.20)
PT −σ M
where the integral is over all Brownian curves {γτ0 = (xτ0 , T − σ − τ)}τ∈[0,T −σ] based at γσ = (xσ , T − σ) with
respect to the measure Γγσ , and γ|[0,σ] ∗ γ0 ∈ P(x,T ) M denotes the concatenation of γ|[0,σ] and γ0 .
Proof. Using Proposition 3.7 we see that the martingale problem for (3.6) is well posed. Thus, by the
Stroock-Varadhan principle, c.f. [SV79, Thm. 10.1.1], we have the strong Markov-property
Uu
u
Eu [ f (Uσ+τ
)|Σσ ] = EUσu [ f (Uτ σ )]
(3.21)
for all test functions f : F → R and all stopping times σ ≤ T , where {Uτu0 }τ∈[0,π2 (u0 )] denotes the solution of
(3.6) with initial condition u0 . Pushing forward via π : F → M, and choosing σ constant, equation (3.21)
implies
"
(x,T )
!#
h (x,T ) i
Xσ ,T −σ
σ
E(x,T ) f Xσ+τ | Σ = E X (x,T ) ,T −σ f Xτ
(3.22)
σ
for all test functions f : M → R. Note that equation (3.22) is exactly equation (3.20) for the case that F
is the 1-point cylinder function f ◦ uσ+τ .3 Now, if F is a k-point cylinder function, then by conditioning
at the first evaluation time we can split up the computation of its (conditional) expectation to computing an
expectation of a 1-point cylinder function and of a (k − 1)-point cylinder function. Arguing by induction, we
infer that (3.20) holds for all cylinder functions. Since the cylinder functions are dense in the space of all
integrable functions, c.f. Definition 3.18, this proves the proposition.
3
If F = f ◦ uσ0 is a 1-point cylinder function with σ0 ≤ σ, then (3.20) holds true trivially.
15
For any F ∈ L1 (PT M, Γ(x,T ) ), the induced martingale F τ = E(x,T ) [F|Στ ] is defined by taking the conditional expectation with respect to the σ-algebras Στ for every τ ∈ [0, T ]. It indeed has the martingale
property
0
E(x,T ) [F τ |Στ ] = F τ
(τ0 ≥ τ).
(3.23)
The quadratic variation of the martingale F • = {F τ }τ∈[0,T ] (and more generally of any stochastic process
where the following limit exists) is defined by
X
[F • ]τ = lim
(F τk − F τk−1 )2 ,
(3.24)
||{τ j }||→0
k
where the limit is taken in probability, over all partions {τ j } of [0, τ] with mesh going to zero.
Assume now that F ∈ L2 (PT M, Γ(x,T ) ). Then the convergence in (3.24) is not just in probability but also
in L1 . Moreover, we have the Ito isometry
h 0
h
i
i
(3.25)
E [F • ]τ0 − [F • ]τ Στ = E (F τ − F τ )2 Στ .
The differential of [F • ]τ takes the form d[F • ]τ = Yτ dτ for some nonnegative Στ -adapted stochastic process
•
Y, which we denote by Yτ = d[Fdτ ]τ . Using Fatou’s lemma and equation (3.25) it can be estimated by
i
i
1 h •
d[F • ]τ
1 h τ+ε
• τ
τ 2 τ
≤ lim inf
E
[F
]
−
[F
]
E
(F
−
F
)
Σ
=
lim
inf
Σ ,
(3.26)
τ+ε
τ
ε→0+ ε
ε→0+ ε
dτ
for almost every τ for almost every γ.
3.4
Heat equation and Wiener measure
The goal of this section is to explain the relationship between the Wiener measure and the heat equation
on our evolving manifolds. In particular, we will see that the Wiener measure is indeed characterized by
equation (1.9). We start with the following representation formula for solutions of the heat equation.
Proposition 3.27 (Representation formula for solutions of the heat equation). If s ∈ [0, T ], and w is a
solution of the heat equation, ∂t w = ∆gt w, with w| s = f ∈ Cc∞ (M), then w(x, T ) = E(x,T ) [ f (XT −s )].
Proof. By Definition 3.14 we have w(Xτ , T − τ) = w(U
˜ τ ), where w˜ denotes the lift of w to the frame bundle,
which is constant in fibre directions. By the Ito formula (Proposition 3.7) we have
dw(U
˜ τ ) = Hi w(U
˜ τ ) dWτi + Dτ w(U
˜ τ ) dτ + ∆H w(U
˜ τ ) dτ,
(3.28)
P
where ∆H = ni=1 Hi2 is the horizontal Laplacian. Since w solves the heat equation, the sum of the last two
terms vanishes (see Lemma 3.1 and Lemma 3.3), and by integration we obtain
Z T −s
Hi w(U
˜ τ ) dWτi .
(3.29)
w(U
˜ T −s ) − w(U
˜ 0) =
0
Note that w(U
˜ 0 ) = w(u)
˜
= w(x, T ), and that w(U
˜ T −s ) = w(XT −s , s) = f (XT −s ) = f (π1 UT −s ). Moreover, after
taking expectations the term on the right hand side of (3.29) disappears by the martingale property, i.e. since
the integrand is Στ -adapted (c.f. Definition 3.18), and since Brownian motion has zero expectation. Thus,
w(x, T ) = Eu [ f (π1 UT −s )] = E(x,T ) [ f (XT −s )],
(3.30)
as claimed.
16
Proposition 3.31 (Characterization of the Wiener measure). If eσ : P(x,T ) M → M k is the evaluation map at
σ = {0 ≤ σ1 ≤ . . . ≤ σk ≤ T }, given by eσ (γ) = (π1 γσ1 , . . . , π1 γσk ), and if we write si = T − σi , then
eσ,∗ dΓ(x,T ) (y1 , . . . , yk ) = H(x, T |y1 , s1 )dvolgs1 (y1 ) · · · H(yk−1 , sk−1 |yk , sk )dvolgsk (yk ).
(3.32)
Moreover, equation (3.32) uniquely characterizes the Wiener measure on P(x,T ) M.
Proof. By Propositon 3.27 we have the equality
Z
Z
H(x, T |y, s) f (y)dvolg(s) (y) =
M
P(x,T ) M
f (π1 γσ )dΓ(x,T ) (γ)
(3.33)
for every test function f , say smooth with compact support. Since these functions are dense in the space of
all integrable functions on M, this proves (3.32) for k = 1.
Now, if f : M k → R and σ = {0 ≤ σ1 ≤ . . . ≤ σk ≤ T }, then using Proposition 3.19 and what we just
proved, the conditional expectation (e∗σ f )σk−1 = E(x,T ) [e∗σ f |Σσk−1 ] is given by
Z
(e∗σ f )σk−1 (γ) =
f (π1 γσ1 , . . . , π1 γσk−1 , yk )H(π1 γσk−1 , sk−1 |yk , sk )dvolgsk (yk ).
(3.34)
M
Using the formula E(x,T ) [e∗σ f ] = E(x,T ) [E(x,T ) [e∗σ f |Σσk−1 ]] and induction, this proves (3.32).
Finally, by the density of cylinder functions in the space of measurable functions (c.f. Definition 3.18),
equation (3.32) uniquely characterizes the Wiener measure on P(x,T ) M.
3.5
Feynman-Kac formula
We will now prove a Feynman-Kac type formula for vector valued solutions of the heat equation with
potential
∇t Y = ∆gt Y + At Y,
Y| s = Z,
(3.35)
where At ∈ End(T M) is a smooth family of endomorphisms, and Z is say smooth with compact support.
The idea is to generalizes the representation formula for solutions of the heat equation (Proposition 3.27)
in two ways by: i) using stochastic parallel translation (Definition 3.15) to transport everything to T x M, and
ii) multiplication by an endomorphism RT −s = RT −s (γ) : T x M → T x M, which is obtained by solving an
ODE along every Brownian curve γ, to capture how the potential At effects the solution.
Proposition 3.36 (Feynman-Kac formula). If s ∈ [0, T ], At ∈ End(T M), and Y is a vector valued solution
of the heat equation with potential, ∇t Y = ∆gt Y + At Y, with Y| s = Z ∈ Cc∞ (T M), then
Y(x, T ) = E(x,T ) [RT −s PT −s Z(XT −s )],
where Rτ = Rτ (γ) : T x M → T x M is the solution of the ODE
d
dτ Rτ
= Rτ Pτ AT −τ P−1
τ with R0 = id.
Remark 3.38. Similar formulas hold for tensor valued solutions of the heat equation with potential.
17
(3.37)
˜
Proof. Let Y˜ : F → Rn , Y(u)
= u−1 Yπu , be the equivariant function associated to Y. Applying the Ito
formula (Proposition 3.7) to each component, we obtain
˜ τ ) = Hi Y(U
˜ τ )dWτi + Dτ Y(U
˜ τ )dτ + ∆H Y(U
˜ τ )dτ = Hi Y(U
˜ τ )dWτi − A˜ T −τ Y(U
˜ τ )dτ,
dY(U
(3.39)
where we lifted equation (3.35) to F using Lemma 3.1 and Lemma 3.3. Let R˜ τ : Rn → Rn be the solution of
d ˜
the ODE dτ
Rτ = R˜ τ A˜ T −τ with R0 = id. Then
˜ τ ) = R˜ τ Hi Y(U
˜ τ )dWτi .
d R˜ τ Y(U
(3.40)
The right hand side disappears after taking expectations, by the martingale property, as in the proof of
Proposition 3.27. Thus,
˜
Y(u)
= Eu [R˜ T −s Y˜ T −s (UT −s )].
(3.41)
Finally, we can translate from Y˜ to Y by computing
˜
Y(x, T ) = uY(u)
= Eu [U0 R˜ T −s U0−1 U0 UT−1−s UT −s Y˜ T −s (UT −s )] = E(x,T ) [RT −s PT −s Z(XT −s )].
(3.42)
Here, we used that Rτ = U0 R˜ τ U0−1 , which can be checked by computing
d
˜ −1
dτ (U 0 Rτ U 0 )
= U0 R˜ τ A˜ T −τ U0−1 = U0 R˜ τ U0−1 U0 Uτ−1 Uτ A˜ T −τ Uτ−1 Uτ U0−1 = U0 R˜ τ U0−1 Pτ AT −τ P−1
τ ,
which shows that Rτ and U0 R˜ τ U0−1 solve the same ODE, and thus must be equal.
3.6
Parallel gradient and Malliavin gradient
Let F : P(x,T ) M → R be a cylinder function. If γ ∈ P(x,T ) M is a continuous curve and V is a right continuous
vector field along γ, then the directional derivative DV F(γ) is well defined as a limit of difference quotients,
namely
F(γV,ε ) − F(γ)
DV F(γ) = lim
,
(3.43)
ε→0
ε
g
where γV,ε = {(xτV,ε , T − τ)}τ∈[0,T ] is the curve in P(x,T ) M defined by xτV,ε = exp xττ (εVτ ).
Definition 3.44 (Parallel gradient). Let σ ∈ [0, T ]. If F : P(x,T ) M → R is a cylinder function, then its
σ-parallel gradient is the unique almost everywhere defined function ∇kσ F : P(x,T ) M → (T x M, gT ), such
that
DV σ F(γ) = h∇kσ F(γ), vi(T x M,gT )
(3.45)
for almost every Brownian curve γ and every v ∈ (T x M, gT ), where V σ = {Vτσ }τ∈[0,T ] is the vector field along
γ given by Vτσ = 0 if τ ∈ [0, σ) and Vτσ = P−1
τ v if τ ∈ [σ, T ].
Explicitly, if F = u ◦ eσ : P(x,T ) M → M k → R, and if we write s j = T − σ j , then it is straightforward to
check that


 X

( j) 
∇kσ F = e∗σ 
Pσ j gradgs j u ,
(3.46)
σ j ≥σ
18
where grad( j) denotes the gradient with respect to the j-th variable, and Pσ j is stochastic parallel transport.
Let H be the Hilbert-space of H 1 -curves {vτ }τ∈[0,T ] in (T x M, gT ) with v0 = 0, equipped with the inner
product
Z T
hv, wiH =
h˙vτ , w˙ τ i(T x M,gT ) dτ.
(3.47)
0
Definition 3.48 (Malliavin gradient). If F : P(x,T ) M → R is a cylinder function, then its Malliavin gradient
is the unique almost everywhere defined function ∇H F : P(x,T ) M → H, such that
DV F(γ) = h∇H F(γ), viH
(3.49)
for every v ∈ H for almost every Brownian curve γ, where V = {P−1
τ vτ }τ∈[0,T ] .
Let us now explain the extension to operators on L2 . This is based on the integration by parts formula
from the appendix (Theorem A.1), which says that the formal adjoint of DV is given by
D∗V G
= −DV G +
T
Z
1
2G
0
d
vτ − Pτ (Ric + 12 ∂t g)P−1
h dτ
τ vτ , dWτ i.
(3.50)
By the Ito isometry and (1.3) we have the estimate
2
Z T
−1
1
d
E(x,T ) h dτ vτ − Pτ (Ric + 2 ∂t g)Pτ vτ , dWτ i ≤ C|v|2H .
0
(3.51)
Using (3.50), (3.51), and the definition of the formal adjoint, we see that if Fn is a sequence of cylinder
functions with Fn → 0 and DV Fn → K in L2 (P(x,T ) M), then (K, G) = 0 for all cylinder functions G,
and thus K = 0. It follows that ∇H can be extended to a closed unbounded operator from L2 (P(x,T ) M)
to L2 (P(x,T ) M, H), with the cylinder functions being a dense subset of the domain. Similarly, ∇kσ can be
extended to a closed unbounded operator from L2 (P(x,T ) M) to L2 (P(x,T ) M, T x M), again with the cylinder
functions being a dense subset of the domain.
3.7
Ornstein-Uhlenbeck operator
The Ornstein-Uhlenbeck operator L = ∇H∗ ∇H is an unbounded operator on L2 (PT M, Γ(x,T ) ) defined by
composing the Malliavin gradient with its adjoint. More generally, there is a family of Ornstein-Uhlenbeck
Rτ
k
operators Lτ1 ,τ2 on L2 (PT M, Γ(x,T ) ) defined by the formula Lτ1 ,τ2 = τ 2 ∇k∗
τ ∇τ dτ, which captures the part of
1
the Laplacian coming form the time range [τ1 , τ2 ]. The next proposition shows in particular that L = L0,T .
Proposition 3.52. If F : PT M → R is a cylinder function, then for almost every curve γ ∈ (PT M, Γ(x,T ) )
we have the formula
Z
T
|∇H F|2 (γ) =
0
19
|∇kτ F|2 (γ) dτ.
(3.53)
Proof. The cylinder function has the form F = u◦eσ : PT M → M k → R. By the definition of the Malliavin
gradient (Definition 3.48), for almost every γ ∈ (PT M, Γ(x,T ) ) we have
Z T
k
X
( j)
d H
d
hvσ j , Pσ j gradgs j u(eσ j γ)i = DV F(γ) = h∇H F(γ), viH =
h dτ
∇ F(γ), dτ
vi dτ.
(3.54)
0
j=1
It follows that
d H
dτ ∇ F(γ)
=
k
X
( j)
1{τ≤σ j } Pσ j gradgs j u(eσ j γ).
(3.55)
j=1
Based on this, writing σ0 = 0, we compute
|∇
H
F|2H (γ)
=
T
Z
0
d H
| dτ
∇ F(γ)|2 dτ
k
k
2 Z T
X
X
(`)
|∇kτ F|2 (γ) dτ,
Pσ` gradgs` u(eσ` γ) =
(σ j − σ j−1 ) =
(3.56)
0
`= j
j=1
where we used that the integrands are piecewise constant. This proves the proposition.
4
Proof of the main theorem
In this section, we prove our main theorem (Theorem 1.22) characterizing solutions of the Ricci flow.
We will prove the implications (R1)⇒(R2)⇒(R3’)⇒(R4)⇒(R5)⇒(R3)⇒(R2’)⇒(R1). Here, (R3’) denotes the (seemingly stronger) statement that for every F ∈ L2 (PT M, Γ(x,T ) ) we have the pointwise estimate
r
h
i
√
d[F • ]τ
0
(R3 )
(γ) ≤ 2 E(x,T ) |∇kτ F| Στ (γ)
dτ
for almost every γ ∈ P(x,T ) M for almost every τ ∈ [0, T ], and (R2’) denotes the (seemingly weaker) statement that for every F ∈ L2 (PT M, ΓT ), we have the gradient estimate
Z
Z
0
2
(R2 )
|∇ x
FdΓ(x,T ) | ≤
|∇k F|2 dΓ(x,T ) .
PT M
PT M
Before delving into the proof, we observe that it suffices to prove the estimates for cylinder functions,
since this implies the general case by approximation. For illustration, let us spell out the approximation
argument for (R2): Let F ∈ L2 (PT M, ΓT ). Let F j be a sequence of cylinder functions that converges to
F in L2 (PT M, ΓT ) and pointwise almost everywhere. By Fubini’s theorem and the dominated convergence
theorem, for a.e. x ∈ M we obtain that lim j→∞ E(x,T ) F 2j = E(x,T ) F 2 < ∞. We can assume that for a.e. x ∈ M
the function F is in the domain of the parallel gradient based at (x, T ) (since otherwise the right hand side of
(R2) is infinite by convention and the estimate holds trivially). Thus, lim j→∞ E(x,T ) |∇k F j | = E(x,T ) |∇k F| < ∞
for a.e. x ∈ M. If we know that (R3) holds for cylinder functions, then we can infer that
Z
Z
lim sup ∇ x
F j dΓ(x,T ) ≤
|∇k F| dΓ(x,T )
(4.1)
j→∞
PT M
PT M
for a.e. x ∈ M. Once we know that the local Lipschitz-bounds (4.1) holds, then passing to a subsequential
limit we can conclude that (R2) holds for F for a.e. x ∈ M.
20
4.1
The gradient estimate
The goal of this section is to prove the implication (R1)⇒(R2). We start with the following theorem for the
gradient of the expectation value.
Theorem 4.2 (Gradient formula). If (M, gt )t∈I is an evolving family of Riemannian manifolds and F :
PT M → R is a cylinder function, then
#
"
Z T
k
d
R
∇
F
dτ
,
(4.3)
gradgT E(x,T ) F = E(x,T ) ∇k F +
dτ τ τ
0
where Rτ = Rτ (γ) : T x M → T x M is the solution of the ODE
d
dτ Rτ
= −Rτ Pτ (Ric + 12 ∂t g)P−1
τ with R0 = id.
Our proof of Theorem 4.2 is by induction on the order of the cylinder function. The main ingredients are
the Feynman-Kac formula for vector valued solutions of the heat equation (Proposition 3.36), the formula for
the conditional expectation value (Proposition 3.19), and the following evolution equation for the gradient.
Proposition 4.4 (Evolution of the gradient). If (M, gt )t∈I is an evolving family of Riemannian manifolds, and
u solves the heat equation, ∂t u = ∆gt u, then its gradient, gradgt u, solves the equation
∇t gradgt u = ∆gt gradgt u − (Ric + 12 ∂t gt )(gradgt u, ·)]gt .
(4.5)
Proof. Using the formula ∂t (g−1 ) = −g−1 (∂t g)g−1 and the definitions of gradgt (u) and ∇t , we compute
∇t gradgt u = gradgt (∂t u) − ∂t gt (gradgt u, ·)]gt + 21 ∂t gt (gradgt u, ·)]gt
= ∆gt gradgt u − (Ric + 21 ∂t gt )(gradgt u, ·)]gt ,
(4.6)
where we used the equation ∂t u = ∆gt u and commuted the Laplacian and the gradient.
Proof of Theorem 4.2. We argue by induction on the order k = |σ| of the cylinder function F = e∗σ u.
If k = 1, then by equation (1.13) the expectation E(x,T ) F is given by integration with respect to the heat
kernel, namely
Z
E(x,T ) F =
u(y)H(x, T |y, s)dvolg(s) (y) = (P sT u)(x),
(4.7)
M
where s = T − σ. On the other hand, by Proposition 4.4 we have the evolution equation
∇t gradgt P st u = ∆gt gradgt P st u − (Ric + 21 ∂t gt )(gradgt P st u),
(4.8)
where we view (Ric + 21 ∂t gt ) as endomorphism (using the metric gt ). We can thus apply the Feynman-Kac
formula (Proposition 3.36), and obtain
(gradgT P sT u)(x) = E(x,T ) [Rσ Pσ (gradgs u)(Xσ )],
(4.9)
d
where Rτ = Rτ (γ) : T x M → T x M is the solution of the ODE dτ
Rτ = −Rτ Pτ (Ric + 12 ∂t g)P−1
τ with R0 = id.
Using the fundamental theorem of calculus and equation (3.46), we can rewrite this as
"
!
#
"
#
Z σ
Z T
k
k
d
d
(gradgT P sT u)(x) = E(x,T ) id +
R
dτ
P
(grad
u)(X
)
=
E
∇
F
+
R
∇
F
dτ
. (4.10)
σ
σ
(x,T )
gs
dτ τ
dτ τ τ
0
0
21
Thus, the gradient formula (4.3) holds true for 1-point cylinder functions.
Now, arguing by induction, let F = e∗σ u be a k-point cylinder function and let si = T − σi . Note that
E(x,T ) F = E(x,T ) E(x,T ) [F|Σσ1 ],
(4.11)
Using Proposition 3.19 we see that G := E(x,T ) [F|Σσ1 ] is a 1-point cylinder function given by G = e∗σ1 w,
w(y) = E(y,s1 ) [u(y, Xσ0 2 −σ1 , . . . , Xσ0 k −σ1 )],
(4.12)
where the expectation is over all Brownian curves starting at (y, T − σ1 ). Note that by equation (4.11) and
the case k = 1 of the gradient formula we have
gradgT E(x,T ) F = gradgT E(x,T )G = E(x,T ) Rσ1 Pσ1 (gradgs w)(Xσ1 ),
1
where Rτ = Rτ (γ) : T x M → T x M is the solution of the ODE
Using the product rule and induction, we compute
u(y, Xσ0 2 −σ1 , . . . , Xσ0 k −σ1 )
(gradgs w)(y) =E(y,s1 ) grad(1)
g1s
1
"
Z
0k
0
0
+ E(y,s1 ) ∇ u(y, Xσ2 −σ1 , . . . , Xσk −σ1 ) +
0
d
dτ Rτ
(4.13)
= −Rτ Pτ (Ric + 12 ∂t g)P−1
τ with R0 = id.
#
T −σ1
0
0
d 0 0k
dτ Rτ ∇τ u(y, Xσ2 −σ1 , . . . , Xσk −σ1 ) dτ
(4.14)
where X 0 and ∇0k denotes Brownian motion and the parallel gradient based at (y, T − σ1 ), and R0τ = R0τ (γ0 ) :
d 0
0
T y M → T y M is the solution of the ODE dτ
Rτ = −R0τ P0τ (Ric + 12 ∂t g)P0−1
τ with R0 = id. Note that
E(y,s1 ) grad(1)
u(y, Xσ0 2 −σ1 , . . . , Xσ0 k −σ1 ) + E(y,s1 ) ∇0k u(y, Xσ0 2 −σ1 , . . . , Xσ0 k −σ1 )
g1
s
=
k
X
j
E(y,s1 ) P0σ j −σ1 (gradgs j u)(Xσ0 1 −σ1 , . . . , Xσ0 k −σ1 ). (4.15)
j=1
Moreover, if γ = γ|[0,σ1 ] ∗ γ0 then Pτ (γ|[0,σ1 ] ∗ γ0 ) = Pσ1 (γ) ◦ P0τ−σ1 (γ0 ) and thus
−1
Pσ1 R0τ−σ1 P−1
σ1 = Rσ1 Rτ
(4.16)
for τ ≥ σ1 , since both sides solve the same ODE with the same initial condition at time σ1 . Putting
everything together, we conclude that
#
"
#
"
Z T
Z T
k
k
k
d
d
R
∇
F
dτ
=
E
∇
F
+
R
∇
F
dτ
,
(4.17)
gradgT E(x,T ) F = E(x,T ) Rσ1 ∇k F +
(x,T )
dτ τ τ
dτ τ τ
σ1
0
where we also used Proposition 3.19, the formula Pσ j (γ|[0,σ1 ] ∗ γ0 ) = Pσ1 (γ) ◦ P0σ j −σ1 (γ0 ), and (3.46).
Proof of (R1)⇒(R2). The gradient formula (Theorem 4.2), together with the above approximation argument, immediately establishes the implication (R1) ⇒ (R2). To see this, just observe that for families of
R
Riemannian manifolds evolving by Ricci flow the time integral in (4.3) vanishes, that |∇ x P M F dΓ x | and
T
R
|gradgT E(x,T ) F| are the same (just in different notation), and that |E(x,T ) ∇k F| ≤ P M |∇k F| dΓ(x,T ) .
T
22
4.2
Regularity of martingales
The goal of this section is to establish the implication (R2)⇒(R3’). For convenience of the reader, we also
prove the (obvious and logically not needed) implication (R3’)⇒(R3). We start with the following formula
for the quadratic variation of a martingale on path space.
Theorem 4.18 (Quadratic variation formula). If (M, gt )t∈I is an evolving family of Riemannian manifolds
and F : P(x,T ) M → R is a cylinder function, then
d[F • ]τ
(γ) = 2|∇y E(y,T −τ) Fγ[0,τ] |2 (π1 γτ )
dτ
for almost every γ ∈ P(x,T ) M, where Fγ[0,τ] : PT −τ M → R is defined by Fγ[0,τ] (γ0 ) = F(γ|[0,τ] ∗ γ0 ).
(4.19)
Proof of Theorem 4.18. Given a cylinder function F = u ◦ eσ : P(x,T ) M → M k → R, and a number
τ ∈ [0, T ], let j be the largest integer such that σ j ≤ τ. By the formula for the conditional expectation
(Proposition 3.19) and the characterization of the Wiener measure (Propositon 3.31), for ε > 0 small enough,
F τ+ε is given by
Z
τ+ε
F (γ) =
u(π1 γσ1 , . . . , π1 γσ j , y j+1 , . . . , yk )dνγτ+ε (y j+1 , s j+1 ) . . . dν(yk−1 ,sk−1 ) (yk , sk ).
(4.20)
We can write this
Z
wε (z) =
M k− j
as F τ+ε
M k− j
= e∗τ+ε wε , where we define wε = wε,γσ1 ,...,γσ j by
u(π1 γσ1 , . . . , π1 γσ j , y j+1 , . . . , yk )dν(z,T −τ−ε) (y j+1 , s j+1 ) . . . dν(yk−1 ,sk−1 ) (yk , sk ).
d[F • ]τ
dτ
is Στ -measurable, we can compute
"
#
h
i
d[F • ]τ
d[F • ]τ τ
1
Σ = lim E(x,T ) (F τ+ε − (F τ+ε )τ )2 Στ ,
(γ) = E(x,T )
+
ε→0 ε
dτ
dτ
Now, since the function
(4.21)
(4.22)
where we also used the martingale property (F τ+ε )τ = F τ and the definition of the quadratic variation, c.f.
Section 3.3. Using again Proposition 3.19 and Propositon 3.31, as well as some rough short time asymptotics
for the heat kernel, we conclude that
!2
Z
Z
1
d[F • ]τ
(γ) = lim+
wε (z) −
wε (ˆz) dνγτ (ˆz, T − τ − ε) dνγτ (z, T − τ − ε) = 2|∇w0 |2(π1 γτ ). (4.23)
ε→0 ε M
dτ
M
Observing that w0 (y) = E(y,T −τ) Fγ[0,τ] , this proves the theorem.
Proof of (R2)⇒(R3’). Let (M, gt )t∈I be a smooth family of Riemannian manifolds such that the gradient
estimate (R2) holds, and let F : P(x,T ) M → R be a cylinder function. Observe that
|∇kτ F|(γ|[0,τ] ∗ γ0 ) = |∇k0 Fγ[0,τ] |(γ0 ).
(4.24)
Now, using Theorem 4.18, the gradient estimate (R2), and (4.24), we compute (for a.e. γ for a.e. τ)
r
r

h
i
 d[F • ]τ τ  √
d[F • ]τ
Σ  = 2E(x,T ) |∇y E(y,T −τ) Fγ[0,τ] |(π1 γτ ) Στ
(γ) = E(x,T ) 
dτ
dτ
h
h
i √
i
√
≤ 2E(x,T ) Eγτ |∇k0 Fγ[0,τ] | Στ = 2E(x,T ) |∇kτ F| Στ , (4.25)
where we also used Proposition 3.19 in the last step. This proves (R3’).
23
Proof of (R3’)⇒(R3). Let F ∈ L2 (PT M, Γ(x,T ) ). Using the assumption (R3’), the Cauchy-Schwarz inequality, and the definition of the conditional expectation, we compute
E(x,T )
h
i2
d[F • ]τ
≤ 2E(x,T ) |∇kτ F|2 .
≤ 2E(x,T ) E(x,T ) |∇kτ F| Στ
dτ
This proves the martingale estimate (R3).
4.3
(4.26)
Log-Sobolev inequality and spectral gap
In this section, we prove the implications (R3’)⇒(R4)⇒(R5).
Proof of (R3’)⇒(R4). Let F : PT M → R be a cylinder function, and let {Gτ }τ∈[0,T ] be the martingale
induced by the function G = F 2 , i.e. Gτ = E(x,T ) [F 2 |Στ ]. Using the Ito formula and the martingale property
we compute
Z τ2
Z τ2
1 d[G• ]τ
τ2
τ2
τ1
τ1
τ
τ
E(x,T ) [G log G − G log G ] = E(x,T )
d(G log G ) = E(x,T )
dτ.
(4.27)
τ
dτ
τ1
τ1 2G
By assumption (R3’), the Cauchy-Schwarz inequality, and the definition of Gτ , we have the estimate
h
h
i2
i
d[G• ]τ
≤ 2 E(x,T ) |2F∇kτ F| Στ
≤ 8Gτ E(x,T ) |∇kτ F|2 Στ .
dτ
(4.28)
Combining (4.27) and (4.28) we conclude that
E(x,T ) [Gτ2 log Gτ2 − Gτ1 log Gτ1 ] ≤ 4E(x,T )
Z
τ2
τ1
h
i
E(x,T ) |∇kτ F|2 Στ dτ = 4E(x,T ) hF, Lτ1 ,τ2 Fi,
where we used Propositon 3.52 in the last step. This proves the log-Sobolev inequality (R4).
(4.29)
Proof of (R4)⇒(R5). Applying the log-Sobolev inequality for F 2 = 1 + εG and using approximation, we
obtain
E(x,T ) [(Gτ2 )2 − (Gτ1 )2 ] ≤ 2E(x,T ) hG, Lτ1 ,τ2 Gi.
(4.30)
Observing that E(x,T ) [(Gτ2 )2 − (Gτ1 )2 ] = E(x,T ) [(Gτ2 − Gτ1 )2 ], this proves the spectral gap.
4.4
Conclusion of the argument
The goal of this final section is to prove the remaining implications (R5)⇒(R3)⇒(R2’)⇒(R1).
Proof of (R5)⇒(R3). Using the formula for the Malliavin gradient (Proposition 3.52) we can rewrite the
spectral gap estimate (R5) in the form
Z τ2
τ2
τ1 2
E(x,T ) (F − F ) ≤ 2E(x,T )
|∇kτ F|2 dτ.
(4.31)
τ1
Dividing both sides by τ2 − τ1 and limiting τ2 → τ1 we obtain
E(x,T )
d[F • ]τ
≤ 2E(x,T ) |∇kτ F|2 ,
dτ
(4.32)
which is exactly the martingale estimate (R3).
24
Proof of (R3)⇒(R2’). The quadratic variation formula (Theorem 4.18) at τ = 0 reads
|∇ x E(x,T ) F|2 = 12 E(x,T )
d[F • ]τ
|τ=0 .
dτ
(4.33)
Together with the martingale estimate (R3) at τ = 0 this implies
|∇ x E(x,T ) F|2 ≤ E(x,T ) |∇k F|2 ,
(4.34)
which is exactly the gradient estimate (R2’).
Proof of (R2’)⇒(R1). Let (M, gt )t∈I be an evolving family of Riemannian manifolds satisfying the gradient
estimate (R2’). Plugging in a 1-point cylinder function F = u ◦ eσ : PT M → M → R, the estimate (R2’)
reduces to the estimate
|∇P sT u|2 ≤ P sT |∇u|2 ,
(4.35)
c.f. Remark 1.12. Thus, by Theorem 1.5 (only the implication (S3)⇒(S1) is needed), (M, gt )t∈I is a supersolution of the Ricci flow. To show that (M, gt )t∈I is also a subsolution, we will analyze the gradient
estimate (R2’) for a carefully chosen family of 2-point cylinder functions. Namely, given a point (x, T ) ∈ M
in space-time (T > 0) and a unit tangent vector v ∈ (T x M, gT ) we choose a test function u : M × M → R
such that
grad(1)
grad(2)
HessgT u = 0
at (x, x).
(4.36)
gT u = 2v,
gT u = −v,
We consider the 1-parameter family of test functions
F σ (γ) = u(e0 (γ), eσ (γ)),
(4.37)
where σ ∈ [0, T ]. We will now analyze the asymptotics for σ → 0. We start with the rough estimate
E(x,T ) |∇k F σ − v| = O(σ).
(4.38)
Together with the gradient formula (Theorem 4.2) this implies that
lim |gradgT E(x,T ) F σ |2 = 1 = lim E(x,T ) |∇k F σ |2 .
σ→0
σ→0
(4.39)
To compute the next order term, we first note that the gradient formula (Theorem 4.2) yields the estimate
gradgT E(x,T ) F σ = E(x,T ) [∇k F σ ] + σ(Ric + 21 ∂t g)(x,T ) (v) + o(σ).
(4.40)
Using this, we compute
D
E
grad E
σ 2
k σ2
σ
k σ
1 d
d
F
E
F
−
E
[∇
F
]
|
−
E
|∇
F
|
=
v,
|
grad
σ=0
(x,T
)
(x,T
)
σ=0
(x,T
)
(x,T
)
gT
gT
2 dσ
dσ
= (Ric + 21 ∂t g)(x,T ) (v, v).
(4.41)
Together with (4.39), since the gradient estimate (R2’) holds by assumption, we conclude that
(Ric + 12 ∂t g)(x,T ) (v, v) ≤ 0.
(4.42)
Since (x, T ) and v are arbitrary, this proves that (M, gt )t∈I is a subsolution of the Ricci flow. Recalling that
we already know that (M, gt )t∈I is a supersolution of the Ricci flow, this finishes the proof.
25
A
A variant of Driver’s integration by parts formula
The purpose of this appendix is to prove Theorem A.1, a variant of Driver’s integration by parts formula
[Dri92]. We write (F, G) = E(x,T ) FG. Moreover, if vτ ∈ T x M we use the notation hvτ , dWτ i = (U0−1 vτ )i dWτi .
Theorem A.1 (Integration by parts). Let F, G : PT M → R be cylinder functions, let {vτ }τ∈[0,T ] ∈ H, and
write V = {P−1
τ vτ }τ∈[0,T ] . Then
Z T
d
D∗V G = −DV G + 21 G h dτ
vτ − Pτ (Ric + 12 ∂t g)P−1
τ vτ , dWτ i
(A.2)
0
satisfies (DV F, G) = (F, D∗V G).
Proof. We adapt the proof from [Hsu02, Sec. 8] to our setting of evolving manifolds.
Since DV satisfies the product rule it is enough to show that
E(x,T ) [DV F] =
T
Z
1
2 E (x,T ) [F
0
d
vτ − Pτ (Ric + 12 ∂t g)P−1
h dτ
τ vτ , dWτ i]
(A.3)
for all cylinder functions F. We prove this by induction on the order k of the cylinder function F.
k = 1: Let F = e∗σ u be a 1-point cylinder function, and let s = T − σ. Since w(x, t) := P st u(x) satisfies the
heat equation, its gradient satisfies the equation
∇t gradgtw = ∆gt gradgtw − (Ric + 21 ∂t g)(gradgtw, ·)]gt ,
(A.4)
c.f. the proof of Proposition 4.4. By the Feynman-Kac formula (Proposition 3.36) we have
gradgTw (x, T ) = E(x,T ) [Rσ Pσ gradgsu (Xσ )],
(A.5)
d
Rτ = Rτ Pτ (Ric + 12 ∂t g)T −τ P−1
where Rτ = Rτ (γ) : (T x M, gT ) → (T x M, gT ) solves the ODE dτ
τ with R0 = id,
1
and where we view (Ric + 2 ∂t g)T −τ as endomorphism of T M (using the metric gT −τ ).
By equation (3.29) we have
Z σ
u(Xσ ) = w(x, T ) +
∇H w˜ (Uτ ) · dWτ ,
(A.6)
0
where w˜ is the invariant lift of w and ∇H w˜ = (H1 w,
˜ . . . , Hn w)
˜ is its horizontal gradient.
Let {zτ }τ∈[0,T ] ∈ H. Using the above and the Ito isometry, we compute the following expectation value:
Z σ
Z σ
Z σ
†
H
E(x,T ) u(Xσ ) hRτ z˙τ , dWτ i = E(x,T )
∇ w˜ (Uτ ) · dWτ
hR†τ z˙τ , dWτ i
(A.7)
0
0
0
Z
σ
hR†τ z˙τ , ∇H w(U
˜ τ )i dτ
(A.8)
= 2E(x,T )
0
Z σ
h˙zτ , Rτ U0 ∇H w(U
˜ τ )igT dτ.
(A.9)
= 2E(x,T )
0
26
Let Nτ := Rτ U0 ∇H w(U
˜ τ ) = Rτ Pτ gradgT −τ
w(Xτ , T − τ). Integration by parts gives
Z σ
Z σ
E(x,T )
h˙zτ , Nτ igT dτ = E(x,T ) [hzσ , Nσ igT −
hzτ , dNτ igT ] = E(x,T ) hzσ , Nσ igT ,
0
(A.10)
0
where in the last step we used that Nτ is a martingale, c.f. equation (3.40). Putting things together, and
taking also into account that
Z T
Z T
†
E(x,T ) u(Xσ ) hRτ z˙τ , dWτ i = E(x,T ) u(Xσ ) E(x,T ) hR†τ z˙τ , dWτ i = 0,
(A.11)
σ
we obtain
σ
Z T
E(x,T ) u(Xσ ) hR†τ z˙τ , dWτ i = 2E(x,T ) hR†σ zσ , Pσ gradgsu(Xσ )igT .
(A.12)
0
Finally, we let vτ = R†τ zτ . Then
R†τ z˙τ = v˙ τ − Pτ (Ric + 21 ∂t g)P−1
τ vτ ,
(A.13)
and equation (A.3) follows.
k − 1 → k: Let F = e∗σ f be a k-point cylinder function and let si = T − σi . Define a new function of k − 1
variables by
g(x1 , . . . , xk−1 ) = E(xk−1 ,sk−1 ) f (x1 , . . . , xk−1 , Xσ0 k −σk−1 ),
(A.14)
where X 0 is based at xk−1 . Let G : P(x,T ) M → R be the (k − 1)-point cylinder function
G(γ) = g(eσ1 γ, . . . , eσk−1 γ).
(A.15)
In belows computation we will frequently use the Markov property (Proposition 3.19).
The first step is to express
E(x,T ) DV F =
k
X
( j)
E(x,T ) hvσ j , Pσ j gradgs j f (Xσ1 , . . . , Xσk )igT
(A.16)
j=1
in terms of G. To this end, note that for j = 1, . . . , k − 2 we simply have
( j)
( j)
gradgs j g(x1 , . . . , xk−1 ) = E(xk−1 ,sk−1 ) gradgs j f (x1 , . . . , xk−1 , Xσ0 k −σk−1 ).
(A.17)
For j = k − 1 using the product rule and the gradient formula (A.5) we have
(k−1)
0
grad(k−1)
g sk−1 g(x1 , . . . , xk−1 ) = E (xk−1 ,sk−1 ) gradg sk−1 f (x1 , . . . , xk−1 , Xσk −σk−1 )
+ E(xk−1 ,sk−1 ) R0σk −σk−1 P0σk −σk−1 gradgs f (x1 , . . . , xk−1 , Xσ0 k −σk−1 ), (A.18)
k
where R0τ = R0τ (γ) : (T xk−1 M, g sk−1 ) → (T xk−1 M, g sk−1 ) solves the ODE
R0 = id. Taking expectations, we thus obtain
d 0
dτ Rτ
= R0τ P0τ (Ric + 21 ∂t g) sk−1 −τ P0−1
τ with
E(x,T ) DV F = E(x,T ) DV G + E(x,T ) hvσk , Pσk grad(k)
g sk f (Xσ1 , . . . , Xσk )igT
0
− E(x,T ) E(Xσk−1 ,sk−1 ) hvσk−1 , Pσk−1 R0σk −σk−1 P0σk −σk−1 grad(k)
g sk f (Xσ1 , . . . , Xσk−1 , Xσk −σk−1 )igT . (A.19)
27
By the induction hypothesis we have
E(x,T ) DV G =
σk−1
Z
1
2 E (x,T ) [G
0
d
h dτ
vτ − Pτ (Ric + 12 ∂t g)P−1
τ vτ , dWτ i].
(A.20)
Conditioning, using the induction hypothesis for 1-point functions, and unconditioning again, we compute
E(x,T ) hvσk − vσk−1 , Pσk grad(k)
g sk f (Xσ1 , . . . , Xσk )igT
(k)
0
0
= E(x,T ) E(Xσk−1 ,sk−1 ) hP−1
σk−1 (vσk − vσk−1 ), Pσk −σk−1 gradg sk f (Xσ1 , . . . , Xσk−1 , Xσk )ig sk−1
Z T
1
d
h dτ
vτ − Pτ (Ric + 12 ∂t g)P−1
= 2 E(x,T ) [F
τ (vτ − vσk−1 ), dWτ i].
(A.21)
sk−1
Finally, using the induction hypothesis for 1-point functions and the ODE for R0 we compute
0
E(x,T ) E(Xσk−1 ,sk−1 ) hvσk−1 , (Pσk − Pσk−1 R0σk −σk−1 P0σk −σk−1 )grad(k)
g sk f (Xσ1 , . . . , Xσk−1 , Xσk −σk−1 )igT
(k)
−1
0
0
= E(x,T ) E(Xσk−1 ,sk−1 ) h(I − R0†
σk −σk−1 )Pσk−1 vσk−1 , Pσk −σk−1 gradg sk f (Xσ1 , . . . , Xσk−1 , Xσk −σk−1 )ig sk−1
Z sk
= E(x,T ) [F
hPτ (Ric + 12 ∂t g)P−1
(A.22)
τ vσk−1 , dWτ i].
sk−1
Adding (A.20), (A.21) and (A.22) we conclude that
Z T
d
1
vτ − Pτ (Ric + 12 ∂t g)P−1
E(x,T ) DV F = 2 E(x,T ) [F h dτ
τ vτ , dWτ i].
(A.23)
0
This proves the theorem.
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ROBERT H ASLHOFER , C OURANT I NSTITUTE OF M ATHEMATICAL S CIENCES , N EW YORK U NIVER SITY, 251 M ERCER S TREET, N EW YORK , NY 10012, USA
A ARON NABER , D EPARTMENT OF M ATHEMATICS , N ORTHWESTERN U NIVERSITY, 2033 S HERIDAN
ROAD , E VANSTON , IL 60208, USA
E-mail: [email protected], [email protected]
31